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Questions tagged [complex-dynamics]

Complex dynamics is the study of dynamical systems of functions over complex numbers.

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Area of Mandelbrot set: Uniform convergence in Laurent series method

I am reading Erwing and Scholar's paper on analytically computing the area of the Mandelbrot set and I hope, in a shred of such, that someone might have an idea why swapping an integral and sum is ...
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Boundary of basin of attraction of $\infty$ = closure of repelling periodic points.

I have met five different definitions of the Julia set and I am trying to work out why they are equivalent. I haven't managed to find a reference showing why two of these are equivalent. Why is the ...
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Preimage of circle with critical point and its image inside is simple and closed

I am reading Devaney's "An Introduction to Chaotic Dynamical Systems" and I am trying to convince myself of a claim made there (Section 3.6, proposition 6.2). The proposition concerns showing that the ...
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How to compute expansion factors for hyperbolic rational maps?

It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...
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Mandelbrot set perturbation theory: When do I use it?

I have read the post on Perturbation of Mandelbrot set fractal. I will also be referring to the PDF by K.I. Martin on this topic. My question is to do with the precision laid out in the Martin paper. ...
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Equipotential curves of a Julia set

In 'Dynamics in One Complex Variable' is states that a polynomial $f$ of degree $n$ maps the equipotential $G^{-1}(c) = \{z; G(z)=c\}$ to $G^{-1}(nc)$. I have been thinking about this and I can not ...
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Existence of fixed simple closed curve by polynomials

As the problem mentioned in the title, I wonder that if there exists a simple closed curve on the complex plane which is not circle that can be fixed by a non-linear polynomial with complex ...
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1answer
135 views

Perturbation theory to speed up Julia fractal drawing

I have a really underpowered platform here and want to draw a Julia (and possibly Mandelbrot and Burning ship) fractals using a 8.24 fixed point class. I use iteration count for coloring and need to ...
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Infinite Periodic Points for Rational Functions [closed]

I'm attempting to come up with a proof that the Julia Set of a rational function is not empty. If I could prove that rational functions have infinitly many periodic points, I would be done. However, I ...
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Are there any unique places in the Mandelbrot Set that have not yet been seen graphically?

I'm aware the the Mandelbrot Set is an infinite set of numbers. And that there are many beautiful patterns to be found in it. So my question is are there any more beautiful patterns that we have yet ...
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Difference between limbs and bulbs in Mandelbrot Set

Taking a look to the picture of the Mandelbrot set, one immediately notice its biggest component which we call the main cardioid. This region is composed by the parameters $c$ for which $p_c$ is ...
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Mandelbrot set reference to get started

I would like to learn seriously the mandelbrot set. The idea is to handle it well enough to see why proving its locally connectedness is so difficult. Can you suggest me some books/PDF online? I know ...
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Given a complex number $c_1$, is there a theorem, or any other way that guarantees that $c_1$ belongs to at least one Julia set?

Let $c_1\in \mathbb{C}$. Let $f_{c_2}(z)=z^2+c_2$ be a quadratic polynomial with $c_2\in\mathbb{C}$, and let be the Julia set defined as $$J(f_{c_2})=\{ z\in\mathbb{C}:\forall n\in\mathbb{N}, |f_{c_2}...
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Julia set equality proof

I'm following the book Fractal Geometry, by K. Falconer, and in Chapter 14, Theorem 14.10 he proves that $J(f)=J_0(f)$, for $f $ polynomial and $J(f) = $ {closure of the set of repelling periodic ...
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1answer
67 views

Counterexample of Julia Set definition

So I'm following a book by Falconer on Fractals, and it defines the Julia set of a complex function as $$J(f)=\overline{\{z\in\mathbb{C}:z \text{ is a repulsive periodic point of }f\}}$$ Where the ...
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correctness of Mandelbrot set distance estimation rendering method

I came up with an algorithm that seems to work well in practice for (interior and exterior) distance estimate rendering of the Mandelbrot set, for each starting point $c$: $d := 0$ $z := 0$ $m := \...
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The height of the Mandelbrot Set [duplicate]

A simple question, but one I am unable to find the answer to; What is $$\sup\{y:x+y\cdot i \in M\}$$ Where $M$ is the Mandelbrot set. How is this constant calculated? Is it a known constant? Etc...
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bounds on size of Julia sets inside Mandelbrot set

Let $J_c$ be the Julia set for the quadratic polynomial $f_c(z) = z^2 + c$, and the Mandelbrot set is $M = \{ c \in \mathbb{C} : J_c \text{ is connected} \}$. Call the closed disc of radius $2$ ...
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are parabolic points in the Mandelbrot set algebraic numbers?

Define the iterated quadratic polynomial: $$ \begin{aligned} f_c^0(z) &= 0 \\ f_c^{n+1}(z) &= f_c^n(z)^2+c \end{aligned} $$ The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot ...
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1answer
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counting Misiurewicz points

I enumerated the number of Misiurewicz points using SageMath to factor into irreducible polynomials over $\mathbb{Z}$, where the degree (after discarding factors corresponding to lower (pre)periods) ...
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How does the definition of Mandelbrot set rely on the starting point of the iteration?

Let $f_c(z)=z^2+c$. The Mandelbrot set is the subset of the complex plane given by $$M=\{c\in\mathbb{C}:\exists s\in\mathbb{R}^+,\forall n\in\mathbb{N},|f_c^{(n)}(0)|\leqslant s\},$$ where $f_c^{(...
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How to compute gradient of complicated scalar function ( limit and iteration)?

I have a function which gives scalar potential: $$P(c) = \lim_{n \to \infty} \frac{1}{2^n} \ln|f^{n}_c(0)|$$ where: $c$ is complex variable $f$ is the complex quadratic polynomial $$f_c(z) = z^2 ...
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On a theorem for determining what is not in the filled-in Julia set of a function

Let $f(z) = z^2 + c$ for some $c\in\Bbb C$, and let $R=\dfrac{1+\sqrt{1+4\lvert c\rvert}}2$. If for some $a\in\Bbb C$ and $n>0$ we have that $\lvert f^n(a)\rvert>R$, then $f^n(a)\to\infty$ as $n\...
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is it always possible to choose a small enough positive $\varepsilon$ such that $0 < c_n (\varepsilon) < 1$?

Let $ Y_{n, m} (t) = \left\{ \begin{array}{ll} t & m = 0\\ t + h_{n, m} \cos (\pi n) \tanh \left( \frac{Z (Y_{n, m - 1} (t))}{| \Omega (t) | \prod_{k = 1}^{n - 1} \tanh (Y_{n,...
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71 views

How to solve $ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $?

How to Find analytic $f(z)$ such that $$ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $$ Koenigs function can not be used here So I am stuck. How does the riemann surface look like ?
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Parameter-Plane Dynamics of Fixed Points and Their Preimages For Standard Quadratic Julia Sets

I've been using Unity3d (links to YouTube videos I've made are at the end of the post) and taking advantage of pixel shaders to explore (in real-time) the fixed-points of the complex quadratic map: $...
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1answer
49 views

Are the laminations of Julia sets of the same period the same?

For $q_c = z^2+c$ with $c\in \mathcal{M}$, the Mandelbrot set, if $c_1$ and $c_2$ are in the same bulb, will the Julia sets have the same lamination. So, for example, will all Douady rabbits have the ...
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Are limits points continuous on the Fatou set?

In his mémoir on what's now called the Julia set, Julia remarked (translated from French): ...in any region $D$ containing no point of $E'$ [the Julia set], the sequence of $\phi_i(z)$ [the ...
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249 views

Golden spirals in the Mandelbrot set?

The Mandelbrot set is defined by iterations of $f_c(z) = z^2 + c$. When plotted in the parameter plane, images coloured by various methods are full of logarithmic spirals, which occur due to the ...
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Question about link between Julia set and Mandelbrot Set

I'm searching for a rigorous proof that $c$ is in the Mandelbrot set $M$ if and only if its Julia set $J_c$ is connected. I've read this question and answer , I understand the answer that is given ...
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1answer
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Two topological properties in Renormalization Theory

I am using McMullen-Complex Dynamics and Renormalization (p. 99). There you can find the definitions of renormalization and polynomial-like mapping. Assume $f(z) := z^2 + c$, where $c\in\mathbb{C}$ ...
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Quasiregularity almost everywhere (removability)

The three equivalent definitions of quasiregular mapping that I am using are these ones: Let $U\subset\mathbb{C}$ be an open set and $K < \infty$. Then: A mapping $g:U\to\mathbb{C}$ is $...
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1answer
167 views

Fixed point at infinity

I have three questions about a fixed point at infinity. How can be proved the following result? (if it is true) If we have an entire map of degree $d\geq 2$ (i.e. such that the cardinality of ...
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Image of basin of attraction is not an entire Fatou component.

The function $E(z)=\lambda e^{z}$ with $0<\lambda<1/e$ has an attracting point in $p\in \mathbb{R}$. Using Koenings coordinates, if $U$ is the immediate basin of attraction for $p$, why is $E(U)$...
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1answer
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Julia set property: $J(f) = J(f^p)$

Given an holomorphic function $f:\mathbb{C}\to\mathbb{C}$, I got stuck trying to prove the equality of both Julia sets: $J(f) = J(f^p)$ for every $p\in\mathbb{N}$. $J(f^p)\subset J(f)$ is easy and I ...
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How does this algorithm get the limit set of “kissing” Schottky group?

I'm having difficulty understanding why the algorithm presented in this paper works. If I understand correctly, to construct a Schottky group start with $2n$ circles: $A_1...A_n$ and $B_1...B_n$, ...
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1answer
101 views

bounds on dimension of Julia sets inside Mandelbrot set

$\dim_H J(f_c) \ge 1$ for $c \in M$ by connectedness and uncountability of $J(f_c)$. For which points is there equality? $c=0$ and $c=-2$ for sure, but is this an exhaustive list? Notation: $\dim_H$...
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Help understanding a proof on why the Mandelbrot set is fractal

I was reading an excellent answer that was given on Quora (found here) that trys to explain why the Mandelbrot contains smaller approximate copies of itself. Here's the section that I'm interested in: ...
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1answer
62 views

Iteration of mapping with nested iterates in logarithm

Consider the mapping $$ g: [0, 1) \longrightarrow \mathbb{R}, x \longmapsto g(x) := \frac{x}{1-\ln|x|} $$ It is easily seen that $g$ is a bijective continuous mapping from $[0, 1)$ onto itself with $g(...
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How to find the set of $c$ for which the Julia set of $x^2+c$ completely lies in $\mathbb{R}$?

How to find the set of $c$ for which the Julia set of $x^2+c$ completely lies in $\mathbb{R}$? I know that $c=-2$ must satisfies this because $J(x^2-2)=[-2,2]\in \mathbb{R}$. However, for other $c$, ...
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1answer
71 views

Hausdorff Dimension of Julia set of $z^2+2$?

How to find the Hausdorff Dimension of the Julia set of $z^2+2$, which is $J=J(z^2+2)$? I am recently doing Fractal Geometry from a book by Kenneth Falconer. The book gives some estimation about the ...
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Is this proof of Milnor's lemma valid? the one about Newtons method and super-attractive fixed-points corresponding to simple roots

(Milnor's Lemma) Every simple root of $f (t)$ is a super-attractive fixed-point of $N_f (t)$ where $N_f (t) = t - \frac{f (t)}{\dot{f} (t)}$ since a superattractive fixed-point is one ...
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1answer
99 views

Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$

Consider the following iterations : $x_0 = z$ Where $z$ is complex. $x_n = \frac{ x_{n-1}^2 - 1}{n}$ It is well known that for real $z > 3$ the sequence grows double exponentially. It is known ...
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1answer
43 views

effective degree for normalized escape-time of hybrids

In Renormalizing the Mandelbrot Escape, Linas Vepstas derives the normalized (smooth, continuous) escape-time formula for the exterior of the Mandelbrot set. On the page is also written: Iterating ...
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Is the Mandelbrot set path-connected?

The Mandelbrot set is known to be connected but whether it is path-connected is an open question. But what is the general consensus/belief among mathematicians? I am unable to convince myself either ...
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finding the periods of miniships in the Burning Ship

The Burning Ship fractal is similar to the Mandelbrot set, only instead of being defined by a complex quadratic polynomial, it can defined by two real functions: $$\begin{aligned}X &\leftarrow X^2 ...
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69 views

Do two exponential spirals intersect?

I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), .....
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fast algorithms for external angle computations

Two related problems related to the complex quadratic polynomial $f_c(z) = z^2 + c$ and Mandlebrot and/or Julia sets: find an external angle $\theta_c$ for a complex point $c$ find a complex point $...
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2answers
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Superattracting period-n orbits in the Mandelbrot set

So I'm trying to find the number of superattracting period-$n$ orbits in the family $z\rightarrow z^2+c$ for $n = 1,2,3,4,5,6$. I think I found an algorithm to compute this. $0 \rightarrow c \...
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Shortcut to $x\uparrow \uparrow n$ for very large $n$ and $x\approx e^{(e^{-1})}$?

If the number $x$ is very close to $e^{(e^{-1})}$ , but a bit larger, for example $x=e^{(e^{-1})}+10^{-20}$, then tetrating $x$ many times can still be small. With $x=e^{(e^{-1})}+10^{-20}$ , even $x\...